Differential mobility analyzers (DMAs) are the most powerful instruments available for sizing and classifying particles, especially in the diameter range below 100 nanometers. The most common DMA design involves two concentric cylindrical electrodes. The commercial versions of various instruments have served rather well the aerosol community's for several decades. A number of cylindrical designs have been studied, with different ratios between the electrode radii R1 and R2 and the axial distance L between inlet and outer slits for the aerosol flow. Other geometrical variations upstream the inlet or downstream the outlet slits have been tested for special purposes such as reducing the particle losses or the pressure drop, or for improving flow laminarity at moderate Reynolds number or to reach unusually high Reynolds numbers. As is well known to those skilled in the art, the Reynolds number is a dimensionless number which is equal to the density of a fluid times its velocity times a characteristic length, divided by the fluid's viscosity coefficient.
DMAs are used to separate small charged particles suspended in a gas according to their electrical mobility Z. They combine particle-free fluid flow (the sheath gas) and electric fields to drive charged particles introduced through a first narrow slit (the inlet or injection slit) located in a first electrode into a second narrow slit located in a second electrode. The space between these two slits and electrodes will be referred to as the “working section” of the DMA. Ideally, among the particles introduced through the injection slit, only those with mobilities contained within a relatively small range ΔZ centered about a mean value Z are sampled through the outlet slit. The inverse of the ratio ΔZ/Z is a measure of the DMA resolution. Traditionally, DMAs have been used for the separation of particles considerably larger than 5 nanometers. However, developments over the last decade have made these instruments suitable also for the separation of particles a few nanometers in diameter, and even smaller ions. Their use in the analysis of suspended ions and macroions is therefore of considerable practical interest. Such applications would benefit from the development of instruments of higher resolution and wider range than those that have been traditionally available. It should be noted that the term ion, as used in the instant application, refers not only to molecular ions, but also to charged clusters and in general to any charged particle.
The main obstacle limiting DMA resolution in the nanometer diameter range is Brownian motion. It is known that the associated peak broadening can be reduced considerably by two different means: (a) a geometrical design taking advantage of the existence of an optimal relative positioning between the two slits, and (b) increasing the Reynolds number (Re) of laminar operation of the sheath gas flow in the DMA to values as large as possible. Rosell-Llompart et al., Minimization of the diffusive broadening of ultrafine particles in differential mobility analyzers, in Synthesis and Characterization of Ultrafine Particles, pp. 109–114 (1993), the subject matter of which is herein incorporated by reference in its entirety, discloses high Reynolds number formulation accounting only for Brownian diffusion broadening in cylindrical DMAs. The relative full width ΔZ/Z of the mobility peak associated to particles of fixed mobility Z can be written as(ΔZ/Z)2=16 ln 2D/(L*U)(b+1/b);  (1)b=L/L*;L*2=(R22−R12)2/[2(R22+R12)].  (2,3)D is the diffusivity of the particles, related to their electrical mobility Z viaZ=De/(kT),  (4)where e is the elementary charge (the particles are taken to be singly charged), k Boltzmann's constant and T the absolute temperature. R1 and R2 are the radii of the inner and outer cylindrical electrodes. L is the axial distance between the inlet and outlet slits, and U is the fluid velocity, taken to be independent of the radial coordinate r (plug flow). Suitable generalizations exist of these results for other velocity profiles, plane geometries, and even converging two-dimensional or axisymmetric situations. But equations (1–3) are representative of such broader cases, and suffice for the purposes of the present discussion. Since (b+1/b)≧2, it is clear that, at given radii R1 and R2 and fixed speed U, the resolution is maximized when the length L is equal to L*. The advantages of using DMAs of near-optimal length were first demonstrated experimentally by Rosell-Llompart et al. (1993).
The need to use very high Reynolds numbers follows also immediately from (1). L* coincides with the width Δ=R2−R1 of the working section in the limit of a small gap, Δ<<R2 (when R1 tends to R2), and is reasonably close to it even if R1/R2 differs substantially from unity (L*/Δ=0.843 when R1/R2=0.222). Hence, the ratio D/(L*U) is fairly close to the Peclet number defined here asPe=D/(ΔU).
For the purposes of separating efficiently small species according to their mobility, it is desirable that ΔZ/Z be as small as 1%, even for ions with diameters as small as 1 or 2 nanometers. Since (b+1/b)≧2, this requirement implies that L*U/D>2.22×105. Note also that small ions in standard air have mobilities of 2 cm2/V/s, with associated diffusivities D=0.05 cm2/s. The dimensionless ratio between the kinematic viscosity of air ν(=0.15 cm2/s) and D is therefore ν/D=3, and the quantity UL*/ν (close to the Reynolds number Re=UΔ/ν) needs then to be as high as 0.74×105. We shall see that, in order to cover a wide range of particle sizes, it is convenient to use DMAs with a distance L between the inlet and outlet slits as large as 3L* or even larger (b>3), in which case the resolution is reduced by a factor [(b+1/b)/2]1/2. To compensate for this effect calls for Reynolds numbers (Re) in excess of 105.
The need for high Reynolds (or Peclet) numbers to moderate diffusion in convective diffusive flows is well known. However, the practical exploitation of this knowledge is made difficult by the natural tendency of high Reynolds number flows to become turbulent, as well as by the difficulties associated to the generation of the rather large flows required. For instance, Rosell, J., I. G. Loscertales D. Bingham and J. Fernández de la Mora “Sizing nanoparticles and ions with a short differential mobility analyzer”, J. Aerosol Science, 27, 695–719, 1996, have demonstrated an ability to reach Reynolds numbers as large as 5000 in a variant of the widely used DMA disclosed by Winklmair, et al., A New Electro-mobility Spectrometer for the Measurement of Aerosol Size Distributions in the Size Range from 1 to 1,000 nanometers, J. Aersol Sci., Vol. 22, pp. 289–296 (1991), (commonly referred to as the “Vienna DMA”). But they needed flow rates of some 800 liters/minute, with associated pressure drops close to half an atmosphere. Under such conditions it would have been rather difficult to attain the desired range of Reynolds numbers up to 105.
Some important aspects of the problem of achieving high Reynolds numbers, while avoiding turbulent transition, have been addressed in U.S. Pat. Nos. 5,869,831 and 5,936,242, both to de la Mora, et al., the subject matter of which are herein incorporated by reference in their entirety, following the method of greatly reducing the level of perturbations in the inlet sheath gas flow by means of several stages of laminarizing screens and filters followed by a large contraction which accelerates substantially the sheath gas prior to the working section. For brevity, this large inlet contraction will be referred to as the “trumpet”, even in non-axisymmetric designs.
Some additional clarifications are required here on the various means available to delay transition to relatively high Reynolds numbers. It is well known that fully developed parabolic flow inside a tube tends to become turbulent at a critical Reynolds number near 2000, and that this critical value can be increased greatly when the inlet flow is carefully freed from velocity fluctuations. Often, the velocity profile at the entry of the working section is far closer to flat than parabolic, and this profile is less unstable than the parabolic flow. Still, the boundary layers forming near the cylindrical electrode walls tend also to become turbulent, and the critical conditions at which this happens are also pushed to considerably larger Reynolds numbers by a highly laminar inlet flow. Even so, transition eventually occurs. Furthermore, even in the most carefully prepared laminarizing system, it is very difficult to avoid all external sources of velocity fluctuations. And even when the fluctuation level of the entering flow is very small, local perturbations will tend to appear in the unstable mixing layer following the aerosol inlet. This last difficulty is addressed in the Vienna DMA by a slight reduction in the DMA cross section immediately after the inlet slit, which tends to stabilize the flow. However, this feature is meant to stabilize flows at Reynolds numbers well below 2000, and is likely to be ineffective at Re=105. A recent study of a variant of the Vienna DMA supplied with the very large inlet trumpet introduced in U.S. Pat. No. 5,869,831 observes turbulent transition at Re near 35,000. The boundary layers over their cylindrical electrodes evolve nearly as that over a flat plate, for which comparable conditions for transition are observed in an incoming stream with a velocity fluctuation level of the order of 1%. In contrast, free stream turbulence levels some 100 times smaller are required to achieve critical Reynolds numbers in the range 105–106 in non-converging geometries. These observations indicate that, in planar or cylindrical DMAs, neither the large trumpet inlet proposed in U.S. Pat. No. 5,869,831, nor the slight acceleration used following the inlet slit of the Vienna DMA suffice to create laminar flows in the desired range Re=105.